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On some analytic properties of the atmospheric tomography operator: Non-Uniqueness and reconstructability issues

Ronny Ramlau, Bernadett Stadler

TL;DR

The paper addresses the atmospheric tomography problem in adaptive optics, showing that data from a finite set of guide stars under a limited-angle geometry cannot uniquely determine the turbulent profile above the telescope, formalized via the operator $\mathbf{A}$. It develops a geometric overlap analysis using the overlap function $\omega_l$ and proves non-uniqueness by identifying regions with unit overlap that yield nontrivial nullspaces for $G>1$. The authors also investigate classical reconstruction methods, including $Tikhonov$ regularization, $Landweber$ iteration, and the $Kaczmarz$ method, demonstrating that they generally fail to reconstruct physically meaningful turbulence distributions in this setting. Numerical simulations corroborate the theory, highlighting fundamental ill-posedness and guiding the need for alternative modeling or regularization approaches in MCAO/LTAO/MOAO contexts.

Abstract

In this paper, we consider the atmospheric tomography operator, which describes the effect of turbulent atmospheric layers on incoming planar wavefronts. Given wavefronts from different guide stars, measured at a telescope, the inverse problem consists in the reconstruction of the turbulence above the telescope. We show that the available data is not sufficient to reconstruct the atmosphere uniquely. Additionally, we show that classical regularization methods as Tikhonov regularization or Landweber iteration will always fail to reconstruct a physically meaningful turbulence distribution.

On some analytic properties of the atmospheric tomography operator: Non-Uniqueness and reconstructability issues

TL;DR

The paper addresses the atmospheric tomography problem in adaptive optics, showing that data from a finite set of guide stars under a limited-angle geometry cannot uniquely determine the turbulent profile above the telescope, formalized via the operator . It develops a geometric overlap analysis using the overlap function and proves non-uniqueness by identifying regions with unit overlap that yield nontrivial nullspaces for . The authors also investigate classical reconstruction methods, including regularization, iteration, and the method, demonstrating that they generally fail to reconstruct physically meaningful turbulence distributions in this setting. Numerical simulations corroborate the theory, highlighting fundamental ill-posedness and guiding the need for alternative modeling or regularization approaches in MCAO/LTAO/MOAO contexts.

Abstract

In this paper, we consider the atmospheric tomography operator, which describes the effect of turbulent atmospheric layers on incoming planar wavefronts. Given wavefronts from different guide stars, measured at a telescope, the inverse problem consists in the reconstruction of the turbulence above the telescope. We show that the available data is not sufficient to reconstruct the atmosphere uniquely. Additionally, we show that classical regularization methods as Tikhonov regularization or Landweber iteration will always fail to reconstruct a physically meaningful turbulence distribution.
Paper Structure (4 sections, 4 theorems, 11 equations, 4 figures)

This paper contains 4 sections, 4 theorems, 11 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Setting
  3. Non-Uniqueness of the Atmospheric Tomography Operator
  4. Some geometrical considerations

Key Result

Proposition 2.1

The adjoint of the atmospheric tomography operator ${\bf A}:L_2(\Omega_l)^L \to L_2(\Omega_T)^G$ is given by with

Figures (4)

  • Figure 1: Left: correction of an incoming wavefront by a deformable mirror (image from Au17); Right: sketch of a SCAO system (image from Dhil16).
  • Figure 2: Sketch of an MCAO system assuming a four layer atmosphere and using two deformable mirrors for correction (image from Au17).
  • Figure 3: Layered atmosphere with three viewing directions. At each layer, turbulence can only be reconstructed in the intersections of the shifted pupil areas.
  • Figure :

Theorems & Definitions (6)

  • Proposition 2.1
  • Proposition 2.1
  • Definition 3.1
  • Remark 3.1
  • Proposition 3.1
  • Proposition 3.2